A360878 Number of (undirected) paths in the 2 X n rook graph.

1, 12, 129, 1984, 45945, 1524156, 68838217, 4070403744, 305642504529, 28440008182540, 3214141725643761, 433856895597946272, 68964321078341276809, 12753724616472980432124, 2715405762438952565521785, 659549661987730244294458816, 181293528280954206831103494177

Offset: 1

Author

Andrew Howroyd, Feb 25 2023

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..100

Crossrefs

Row 2 of A360877.

Cf. A096121, A276356, A341500.

Programs

PARI

a(n)={sum(k=2, n, binomial(n,k)*k!) + sum(k=1, n, k*binomial(n,k)*binomial(k-1,k\2)*sum(i=0, n-k, binomial(n-k,i)*(k\2+i)!)*sum(i=0, n-k, binomial(n-k,i)*((k-1)\2+i)!))} \\ Andrew Howroyd, May 28 2025

Formula

a(n) = (Sum_{k=2..n} binomial(n,k)*k!) + (Sum_{k=1..n} k*binomial(n,k)*binomial(k-1, floor(k/2)) * (Sum_{i=0..n-k} binomial(n-k,i)*(floor(k/2)+i)!) * (Sum_{i=0..n-k} binomial(n-k,i)*(floor((k-1)/2)+i)!)). - Andrew Howroyd, May 28 2025

Extensions

a(8) onwards from Andrew Howroyd, May 28 2025